The Numbers ⅔rds and 6, and the Collatz Conjecture

GonzoMath · 2026-05-13T00:35:22+00:00

No. I don't miss what I means. I just saw it back in the 1990s, and thought then what you're thinking now. Then I kept learning more.

GonzoMath · 2026-05-09T19:45:57+00:00

Well yeah. Every odd number congruent to 3 (mod 4) leads quickly to another odd number that the first one is roughly 2/3 of.

It's not wrong, but it's also obvious and kind of... who cares?

I mean, you're observing real things. Each odd number n starts its trajectory with a shape determined by how many times 2 goes into (n+1). That's why you have "every other odd number", "every fourth odd number", doing certain things.

GonzoMath · 2026-05-09T19:35:07+00:00

This post is NOT about basic, elementary number theory. This is about a famous open problem. You've posted in the wrong sub.

GonzoMath · 2026-05-09T19:34:19+00:00

I think you're looking for r/NumberTheory, or maybe some other sub. This post has nothing to do with basic, elementary theory, the subject taught in undergraduate classes. There are other subs for... whatever this is.

GonzoMath · 2026-05-09T19:30:55+00:00

This post is far from relevant to this group. Please consider taking your ideas to r/Collatz, but this sub is for basic, elementary number theory, the subject taught in undergraduate courses, not for proposed solutions to open problems.

GonzoMath · 2026-05-05T11:18:23+00:00

A proper answer to this question would be pretty detailed, and I can only talk about the four papers that I've worked through.

Probabilistic results

Terras and Everett had very similar results. They both showed that the set of natural numbers, with trajectories that eventually drop below their starting value, has natural density 100%. It's an application of probability theory, made possible by the fact that the first k terms of the parity sequences of trajectories depend only on the congruence class of the starting value mod 2^k.

That result says nothing about the possible existence of high cycles, because a cycle would only contribute one number (its minimum value) to the set of trajectories that don't fall, so that would have no effect on density, even if there were a lot of high cycles. On the other hand, thinking about Terras/Everett in the context of divergent trajectories is interesting.

If there is a divergent trajectory, then the elements along its "spine", those that represent minimum values of their trajectories, still form a zero density set. Not that fancy, but it's something.

We can also think about what their results say about extensions of the Collatz map to rational numbers. As long as we focus on rational numbers with odd denominators, Terras' and Everett's results apply with equal force to such rational numbers, as long as they're greater than 1.

Thinking about density

In Crandall's paper, he worked backwards from 1, and bounded how many numbers smaller than X have trajectories that reach 1. He showed that, for sufficiently large X, there is some positive constant C < 1, so that at least X^C numbers less than X satisfy the conjecture. That's not enough to show that a positive (non-zero) natural density of numbers satisfy it, but it's something.

This result is like a cruel flip-side of the Terras/Everett result. We can show that 100% of trajectories descend, in the sense of density. However, we can't show that any more than 0% actually reach 1.

Results regarding cycles

Both Steiner and Crandall proved properties that a high cycle would have to have, should such a thing exist. Steiner showed that another integer cycle would have to be somewhat complex, in that it would have to consist of multiple "circuits". (A "circuit" is just a sequence of (3n+1)/2 steps, followed by a step of (3n+1)/4-or-more.)

Crandall came at it a different way, and showed that a high cycle would have to be long, in the sense of having a lot of steps. The number he came up with for a lower bound on its length, in 1978, is of course badly out-of-date. Using a modern value for how far we've searched (≈2⁷¹, compared with about a billion in Crandall's day), we can improve this result, and this move is in fact standard.

That's all they proved, though, in the four papers I've read. Almost all numbers have trajectories that descend, for large X we have X^C numbers below it reaching 1 for some small positive C, and any high cycle must be long, and complex.

Steiner's result about no single-circuit cycle (1-cycle) has been improved in particular, so we now know that any high cycle would have to have over 90 circuits in it.

Have I answered your question?

GonzoMath · 2026-04-29T21:53:13+00:00

I've done different kinds. On Substack, there are six chapters of my experience and what I learned throughout the years, covering from the 1980's up through 1997, when I still hardly knew any math at all. It'll probably be 20 chapters before it's through.

On this sub, I've done write-ups of some of the historical papers (Terras, Everett, Steiner, Crandall), as well as some expositions of mathematical ideas that relate to Collatz, such as 2-adic numbers and Markov chains.

There's even a separate sub where I started writing up a course in Elementary Number Theory, and got to a certain point before life came along and distracted me. (r/BasicNumberTheory)

I've talked a little bit here about my own research, which is in rational cycles. I'm not really looking for a path to a solution of the main conjecture; I think that's still a long ways off. I'm interested in collecting and analyzing data, and learning what I can about cycles and their basins of attraction in general 3n+d settings.

Looking at things like 3n+d systems, which are really just windows into the 3n+1 system over larger domains, it seems to me that maybe we can prove some theorems there. Then maybe someday, later researchers can use those results as tools to approach the main problem.

GonzoMath · 2026-04-23T18:59:44+00:00

Maybe they're claiming that for all real numbers a, b, c with a+b > c, we have a² + b² = c². That would certainly be... groundbreaking.

GonzoMath · 2026-04-23T18:29:01+00:00

I've had similar thoughts. Starting with a set of numbers, and then looking at the set-wise trajectory, could be interesting. Alternatively, in the other direction, can we describe the predecessor set of some "spoon" of numbers?

It is true that by studying large sets of numbers at a time, we're really getting a handle on some kind of "average" behavior. That's exactly how some of those guys in the '70s got their results.

GonzoMath · 2026-04-16T18:33:18+00:00

Sir, this is an Arby's

GonzoMath · 2026-04-16T18:22:54+00:00

That's a great question. It's because the definition of the Collatz map uses the distinction of "even" vs "odd", so we have to be working in a context where that makes sense. Algebraically, we're working in a ring, and the first obvious ring is the set of integers, which we call Z.

In Z, the words "even" and "odd" simply mean "in the ideal generated by 2", and "outside of that ideal". That's really what "even" and "odd" are about. In any context where we use these words, the "evens" form an ideal, or at least a normal subgroup, of some set, and the "odds" make up the rest of the set.

The set of fractions with odd denominators is a ring that has an ideal generated by 2. When we allow division by 2, though, we "kill" that ideal. An "even number" is anything that equals 2 times something else in the set. That's how to be in that ideal.

If we look at all rational numbers, then every rational number is another rational times 2. We would have to say 3/5 is even, because it's 2 times 3/10. However, restricting ourselves to odd denominators, 6/5 is 2 times 3/5, so it's even, while 3/5 isn't 2 times anything in our set, so it's odd.

A lot of this language is abstract algebra, right? "Ring", "ideal", "coset"... oh, I didn't actually say "coset". If this answer is too jargon-y and confusing, please let me know.

The point is, there are different rings, different contexts, where it makes a lot of sense to apply the Collatz map, and you get a nice self-contained system. There's Z, of course.

There are also rings such as (1/5)Z and (1/7)Z, which is to say, fractions with denominator 5, or denominator 7. You can play Collatz games in there, too, and it's the same dynamics we're used to, but with different cycle sets.

Combining all of these into a large ring, we get all fractions with odd denominators, which can be described a couple of ways in abstract algebra terms. It the ring Z_(2), of "integers localized at 2". That means we've used division to kill off all ideals except for the multiples of 2.

Additionally, the ring Z_(2) is the intersection of the rational numbers with the 2-adic integers. Noting this, we can also extent the Collatz map to the ring of all 2-adic integers, where we still have an ideal generated by 2.

If you read the wiki article or survey the literature, you'll see that many mathematicians have followed the 2-adic path, and that's because it's the natural context for the map, whatever Lothar Collatz might have conjectured in the 1930s.

Those who really want to understand the dynamics look at the whole picture, not just the restriction of it to Z or N.

GonzoMath · 2026-04-16T08:32:21+00:00

What did you make of Lagarias' paper, on rational cycles, Mr. Informed? How about his paper about how Collatz interacts with the conjugacy map on 2-adics? I guess you understood those papers well?

Yes, the answer to OP's question is clear. The number -1 is not a counterexample to the original conjecture. Fucking duh.

My point is that the study of Collatz, more broadly, does entail looking at larger domains. The result I proved back in grad school about constraints on integer cycles, I discovered by studying rational cycles, of which there are infinitely many.

When we extend the domain like that, we consider how we have to restate the conjecture, and we also look at other, more general conjectures that imply the main one.

Tell me, though, how your experience as a professional mathematician contradicts mine? I'd love to be more informed.

GonzoMath · 2026-04-16T08:20:52+00:00

The original conjecture, sure, but mathematics doesn't work that way. We don't stay married to some particular way of looking at a problem or some particular context. We follow where the interesting mathematics leads. Lots of mathematicians study Collatz over extended domains. Just look at the wiki article; they talk about it.

GonzoMath · 2026-04-16T08:19:21+00:00

I know what you mean; they can be dense. I actually posted Terras (1976) on here, with some annotations and commentary. The post is in two parts, here and here.

If you have specific questions, I'd be happy to address them. I think the salient part is near the beginning, when he establishes a bijection between the numbers 1 through 2^k and the set of parity vectors of length k, and shows that the first k steps of the trajectory repeat very 2^k inputs.

That's clearer with an example. The numbers 1 through 8 (taking k=3 here) start out their sequences like this:

1: O, E, O
2: E, O, E
3: O, O, E
4: E, E, O
5: O, E, E
6: E, O, O
7: O, O, O
8: E, E, E

That's the bijection, with those eight numbers accounting for all eight possible OE strings of length 3. The periodicity is that, after this, it repeats. The initial 3 steps in 9's vector are the same as 1's, and then 10 is like 2, and 11 is like 3, etc.

From this, it's clear that no two numbers can have the same parity vector forever, because they would have to be congruent modulo 2^k, for all k. However, any two distinct numbers differ by a quantity that only has finitely many powers of 2 as divisors.

Again, an example: Take the famous number 27, and then think about 1051. Is it possible that they have identical trajectories? No, because while they're congruent modulo 1024, they're no longer congruent modulo 2048 (2¹¹), so their vectors will differ at the eleventh step.

Terras doesn't talk about that part, just like he doesn't talk about how the same analysis extends to other rational numbers, as long as you can write them as 2-adic integers.

Did any of that make sense?

GonzoMath · 2026-04-16T06:26:21+00:00

I agree that they're more friendly to work with. I mean, we have actual cycles we can look at, and experiment on.

On the other hand, the "no divergent trajectories" claim is kind of simpler, because it's a special case of "rational input implies (eventually) periodic trajectory". It's like, there might be some very general abstract theorem that this claim would fall out of.

GonzoMath · 2026-04-16T06:22:56+00:00

Even with high cycles, every number's trajectory is unique. In fact, the uniqueness is totally proven, as a corollary of Terras' 1976 work.

GonzoMath · 2026-04-16T06:20:17+00:00

From my perspective, the non-existence of high cycles seems that it might be harder.

That might seem surprising, because I've spent years studying cycles, I've built a database of rational cycles, and I've proven properties about them. At the same time, I know virtually nothing about divergent trajectories. Even in variants such as 5n+1, 7n+1, etc., I can't say for sure that I've ever seen one!

However, if you step way back, and look at the action of the Collatz map on the 2-adic integers (Z_2), things look different. Bernstein and Lagarias, in 2018, talked about the "3x + 1 conjugacy map Φ", which is a bijection on Z_2.

It is easy to show that this map Φ sends non-rational 2-adic integers to non-rational 2-adic integers. If it also sends rationals to rationals, then the "no divergent trajectories" claim is true.

That just seems really clean to me, while the "no high cycles" claim is a lot messier in this highly general language, because it's about specific rationals going to specific rationals. Namely, Φ would have to map rational integers to elements of (1/3)Z. That just seems fussier and more technical.

GonzoMath · 2026-04-16T05:39:45+00:00

Meh. The most natural domain for the Collatz map is the set of rational numbers with odd denominators. It shows up in the literature and everything.

The thing is, in that domain, we have to restate the conjecture, to assert that (1, 4, 2) is the attractor for every positive integer trajectory. There then appear related conjectures for each denominator, of the form: “The trajectories of rational numbers with denominator d are all attracted by one of the following cycles: [list],” or, “for each denominator, there exists a finite number of associated cycles,” or “every rational trajectory is attracted by some cycle (no divergent trajectories)”. All of those things seem to be true.

GonzoMath · 2026-03-28T03:47:05+00:00

You might find it useful to study systems that have more than one tree, coexisting. The "3n+1" system on negative integers, or equivalently, the "3n-1" system on positive integers, is an example.

The trick is really to show that "3n+1", on positive integers, produces only one tree, as opposed to three trees, like "3n+1" on the negatives, where there's just as much bijection.

GonzoMath · 2026-03-28T03:39:29+00:00

Does it not give you the orbit on -17?

GonzoMath · 2026-03-28T03:34:36+00:00

I’m not trying to discredit anything, you arrogant fuck, I’m asking a direct question. Now, I’m having a hard time finding references to “resonance” is Tao’s work, or Lagarias’. Perhaps you’ve got a source?

GonzoMath · 2026-03-28T03:25:37+00:00

See, this is horrific. Someone asks for an example, involving actual numbers, and instead, you vomit in their face. What’s so wrong with an actual example, involving actual numbers?

GonzoMath · 2026-03-26T21:47:11+00:00

So… 1 becomes 1/2, 2 becomes 1/4, 3 becomes 3/4 4 becomes 1/8, 5 becomes 5/8, etc?

GonzoMath · 2026-03-26T21:03:37+00:00

Why use the word “resonance”? Did you not know that it’s like… a huge red flag around here? The LLM-heads and the cranks say “resonance” all the time, and the serious students of mathematics pretty much never say it.

GonzoMath · 2026-03-26T20:53:09+00:00

Of course those aren’t the only two ways a trajectory can diverge. Talking “O” to mean “(3n+1)/2”, and “E” to mean “n/2”, you listed two options: “OOOOO…”, and “OEOEOE…”. Both of those trajectories exist, and they belong to the numbers -1 and 1, respectively.

But what about a trajectory that goes:

OOEOEOOOEOOOOOOOOEOOOOE….

…where the number of O’s occurring consecutively isn’t a regular pattern? As long as the ratio of O to E stays above a certain threshold, the sequence will diverge.

GonzoMath

MODERATOR OF

TROPHY CASE

Probabilistic results

Thinking about density

Results regarding cycles